Question: Determine how many solutions exist for the system of equations. ${-6x+y = -2}$ ${-12x+2y = -4}$
Solution: Convert both equations to slope-intercept form: ${-6x+y = -2}$ $-6x{+6x} + y = -2{+6x}$ $y = -2+6x$ ${y = 6x-2}$ ${-12x+2y = -4}$ $-12x{+12x} + 2y = -4{+12x}$ $2y = -4+12x$ $y = -2+6x$ ${y = 6x-2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x-2}$ ${y = 6x-2}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-6x+y = -2}$ is also a solution of ${-12x+2y = -4}$, there are infinitely many solutions.